Sparse equidistribution problems in dynamics

26 February - 28 May 2025

Wednesdays, 10:15 - 12:00

Location: HG G 43

First lecture: 26 February

Abstract

Let $(X,T)$ be a topological dynamical system. Classical ergodic theory studies asymptotic behavior of averages of the form $\frac{1}{N}\sum_{n\leq N}f(T^nx)$ where $x\in X$ and $f\in C(X)$. There are many fundamental results on convergence of such averages in various norms: the supremum norm, almost everywhere and in $L^2$ being the main examples (in the latter two we equip $(X,T)$ with a measure $\mu$ which is preserved by $T$). Recently there has been a lot of interest in studying non-classical averages. In the most general form, given a bounded sequence $(a_n)$ one is interested in the asymptotic behavior of averages:

$\frac{1}{\sum_{n\leq N}|a_n|}\sum_{n\leq N}a_n\cdot f(T^nx)$.

In most applications the sequence $(a_n)$ is a structured sequence coming from number theory. Some of the examples of such sequences include:

1. $a_n$= Möbius function or Liouville function (or more generally a multiplicative function).
2. for $k\geq 1$, $a_n=1$ if $n$ is a product of at most $k$ primes, $a_n= 0$ otherwise.
3. $a_n=1$ if $n$ is a k-th power (like square) and $a_n=0$ otherwise.
4. $(a_n)$ a random sequence of real numbers.